Small Additions to Lecture 2

This is a small additions to lecture 2

Perturbing a Linear System
Consider a perturbation to the linear system

$$\begin{cases} \dot{x}=Ax+f(x)\\ x(0)=x_{0} \end{cases} $$

where the perturbation $$ f(x)$$ is non-linear. Then by the theorem of Hartman-Grobman, we can study the local qualitative behaviour of the non-linear system.

The Hartman-Grobman Theorem
According to this theorem, near a hyperbolic equilibrium point $$x(0)=x_{0}$$, the non-linear system

$$ \begin{cases} \dot{x}= f(x)\\ x(0)=x_{0} \end{cases} $$

has the same qualitative structure as the linear system

$$\begin{cases} \dot{x}=Ax\\ x(0)=x_{0} \end{cases} $$ with $$ A = Df(x_{0})$$.